To generate \(B\)  draws, the standard bootstrap with resampling or reweighting requires solving the minimization problem \(\hat{\theta }_m^{(b)} = \text{argmin}_{\theta \in \Theta } Q_m^{(b)}(\theta )\)  \(B\)  times which can be computationally intensive. Faster alternatives have been proposed in the literature. [1]} consider a \(k\) -step nr  update: \(\hat{\theta }_{n,k+1}^{(b)} = \hat{\theta }_{n,k}^{(b)} - [H_n^{(b)}(\hat{\theta }_{n,k}^{(b)})]^{-1}G_n^{(b)}(\hat{\theta }_{n,k}^{(b)})\)  starting at the same full sample estimate \(\hat{\theta }_{n,0}^{(b)} = \hat{\theta }_n\)  for all \(b\) . [2]} shows that \(k=1\)  is enough for asymptotically valid inference, and \(k>1\)  provides asymptotic refinements. dmk  also propose a \(k\) -step quasi-Newton update which requires \(k>1\)  steps.Following earlier work by [3]}, the results are based on quadratic and super-linear convergence properties of nr  and bfgs, respectively, when \(\gamma =1\)  which holds only if \(\Vert \hat{\theta }_{n}^{(b)}-\hat{\theta }_n\Vert \)  is sufficiently small and if (and only if) the quasi-Newton matrix \(P_k\)  satisfies certain properties [4]}. In comparison, using the same \(m=n\) , a \(k=1\) -step implementation of dmk  has the same cost as rnr, but is higher than rnr  for \(k>1\)  steps. The \(k\) -step quasi-Newton update requires \(k>1\)  making it \(k\)  times more costly than rqn  for \(m=n\) . Using \(m<n\)  further reduces the cost of both estimation and inference for rnr, rqn. The original idea of dmk  has been extended to fast subsampling [5]}, two-step estimation [6]}, and \(\ell _1\) -penalized estimation [7]}.
